Introduction

In many count data processes, zero observations occur more frequently than expected from a nominal distribution. Perhaps the most well-known model for such scenarios is the zero-inflated Poisson (ZIP) of Lambert (1992). ZIP can be constructed via two independent latent variables, namely, \(B \sim \mbox{Bernoulli}\) with some probability \(\pi\) of being zero and \(P \sim \mbox{Poisson}\) with some rate \(\lambda\). One desirable feature of ZIP is that the latent Bernoulli construction offers an explicit explanation of the excess zeros. However, the mean of the observed response can only be identified to the product \((1-\pi)\lambda\), using classical ZIP. The goodness-of-fit of ZIP models depends crucially on the individual models for \(\pi\) and \(\lambda\), but this can be not easy to check as neither process is fully observed.

ZIP exponential families

Let \(f(y|\pi, \lambda)\) be the mass function of a classical ZIP family with parameter \(\pi\) & \(\lambda\)). Construct a family \(\{ f_\theta(y); \theta \in \mathbb{R}\}\) of distributions, indexed by \(\theta\), via exponential tilting: \[ f_\theta (y) \propto \exp(\theta y) f(y|\pi, \lambda) \ , \quad \theta \in \mathbb{R}. \]

Each \(f_\theta(y)\) remains a ZIP distribution with new parameters \(\pi_\theta\) and \(\lambda_\theta\) given respectively by \[ \label{eq:1} \pi_\theta = \frac{\pi}{\pi + (1-\pi) e^{\lambda (e^\theta-1)}} \quad \mbox{and} \quad \lambda_\theta = \lambda e^\theta \lambda_\theta = \lambda e^\theta. \]

For mathematical convenience, we set \[\lambda \equiv 1, \quad \nu = \pi/(1-\pi)\quad \text{and}\quad \mu = E(Y_{\theta}) = (1-\pi_\theta) \lambda_\theta. \] We then write the distribution as ZIP\(_{\nu}(\mu)\).

ZIP\(_{\nu}(\mu)\) allows us to construct simple, interpretable regression models via \[ Y|X \sim \mbox{ZIP}_\nu(\mu(X^\top \beta)) \ , \mbox{ for some } \nu \in [0,\infty) \] where \(E(Y|X) = \mu(X^{\top}\beta)\) for some mean function \(\mu(\cdot)\). This is implemented, with model diagnostic tools, in izipr package of Fung and Huang (2021).

bioChemist dataset

The dataset contains the number of articles produced by 915 graduate students in biochemistry during the last three years of their PhD, along with some information on the graduates, such as gender, marital status, the number of kids under 5, how prestigious the department is and mentor’s publication record over the same period.

We fitted the classical ZIP using the zeroinfl() in the pscl package as well as our own glm.izip() in the izipr package.

Interpretating the models

Suppose we want to interpret the effect of kid5, the number of kids under 5.

For the classical ZIP, interpretation requires two-steps. Each additional kid under 5 is associated with an increase in the log-odds of being in the subpopulation that did not have the opportunity to produce a paper of 0.217, which translates to \(\exp(0.217) = 1.242 \approx\) 24% increase in odds.

Given a graduate is in the other subpopulation that have the opportunity to produce paper(s) then each additional kid under 5 is associated with a decrease in the expected number of papers by a factor of \(\exp(\)-0.143) = 0.87, i.e. 13% decrease.

For the ZIP\(_\nu\) model, model interpretation is similar to a log-linaer model. The effect of each additional kid under 5 is a multiplicative factor of \(\exp(\)-0.190) = 0.82, i.e. 18% decrease to the expected number of papers produced. This value has already been adjusted for zero-inflation.

Predicting the zero-inflation

Notice that all variables with a positive effect on the Poisson component of the classical ZIP model had a negative effect on the Bernoulli component. In other words, as the expected number of papers produced increases, the probability of being in the ``do not have opportunity to write a paper" (i.e., zero-inflation) subpopulation tends to decrease, and vice versa.

The strong negative relationship here provides a clear example of how constant zero-inflation can be unrealistic in practice. But this is the assumption used in some time series model for counts in the literature as in Yau, Lee, and Carrivick (2004), Zhu (2012), Yang, Cavanaugh, and Zamba (2015),

ZIP\(_\nu\) for count time-series

ZIP\(_\nu\) distributions prove even more useful for modelling count time-series, as we only need to construct a single ARMA-type recursion for the conditional mean of the process, rather than two latent processes which are only partially observed.

An integer-valued generalized autoregressive conditionally heteroskedastic (INGARCH) time-series model of order \((s, q)\) based on ZIP\(_\nu\) distributions can be specified via \[ \begin{aligned} & Y_t | \mathcal{F}_{t-1} \sim \mbox{ZIP}_\nu(\mu_t) \\ & \mu_t = \delta + \alpha_1 \mu_{t-1} + \ldots \alpha_s \mu_{t-s} + \beta_1 Y_{t-1} + \ldots \beta_q Y_{t-q} \end{aligned} \] where \(\delta, \alpha_1, \ldots, \alpha_s, \beta_1, \ldots, \beta_q > 0\). We call such processes \(\text{ZIP}_\nu\)-INGARCH\((s,q)\). This is also implemented in the izipr package.

If you are interested in what the izipr package can do, please scan the QR-code below.